I've just assumed from the start that the "pee tape" was something that was made up because it became an open secret that Putin had blackmail material and they needed something out there that sounded plausible as blackmail material without screaming "he's fucking evil".

And despite the Epstein files that were released being cherry picked, they still revealed that the actual blackmail material was way worse. My current assumption is that Putin caught wind of what Epstein was up to and KGB'd his way into it. The whole thing (with Russia) has looked like a combination of an adversarial and cooperative relationship. It's possible Epstein was arrested and the whole thing brought to light specifically as an attempt to reduce Putin's power over them (the western elite that Epstein had slowly built up blackmail material on over the years), because before it became public knowledge, he probably just had to bring up that he knows about it and could tell the world to get cooperation from those caught in the web.

Edit: apparently I changed the opening sentence without removing the previous one, removed that now.